Method and system for consolidating phase current samples

ABSTRACT

A method of detecting faults on a power transmission line system includes simultaneously measuring phase current samples at each phase of each transmission terminal; calculating real and imaginary phaselets comprising partial sums of the phase current samples; for each phaselet, calculating a respective partial sum of squares of each phase current sample; calculating the sums of the real and imaginary phaselets over a variable size sliding sample window; calculating real and imaginary phasor components from the phaselets and a sum of the partial sums of the squares over the sample window; using the sums of the real and imaginary phaselets, the real and imaginary phasor components, and the sum of the partial sums of the squares to calculate a sum of squares of errors between the phase current samples and a fitted sine wave representative of the real and imaginary phasor components; using the sum of squares of errors to calculate a variance matrix defining an elliptical uncertainty region; determining whether a disturbance has occurred, and, if so, re-initializing the sample window; and determining whether a sum of current phasors from each terminal for a respective phase falls outside of the elliptical uncertainty region for the respective phase.

This application is a division of application Ser. No. 09/104,760, filedJun. 25, 1998, now U.S. Pat. No. 6,311,307, which is a division of Ser.No. 08/713,295 filed Sep. 13, 1996, now U.S. Pat. No. 5,809,045, whichis hereby incorporated by reference in its entirety.

BACKGROUND

High speed detection of faults on multi-terminal power systemtransmission lines has been attempted by using digital currentdifferential measurements. Differential techniques rely on the factthat, under normal conditions for each phase, the sum of currentsentering terminals equals the charging current for that phase. In oneconventional digital differential current system, the procedure is tocompare individual samples or use a one-cycle window, use a conventionaldual slope operate-restraint characteristic, and compensate for linecharging. This system is not flexible enough to operate in both high andlow bandwidth communication channels. Additionally, the sensitivity ofthis system is low because conventional operate-restraintcharacteristics are not adaptive. In another conventional digitaldifferential current system, charges are calculated at both ends of atwo terminal system by integrating respective current signals and arethen compared. This system has sensitivity limitations and works onlyfor two terminal embodiments.

Many power system monitoring, protection, and control functions could beperformed more efficiently and accurately if power system digitalmeasurements at multiple locations were synchronized. Generally suchmeasurements are only somewhat synchronized because of difficulty inaccurately synchronizing sampling clocks physically separated by largedistances. Conventional uses of digital communications to synchronizesampling clocks at remote locations have accuracies limited byuncertainties in the message delivery time. In particular, digitalcommunications can have different delays in different directions betweena pair of locations which lead to an error in clock synchronization.

Variable size data windows in power system protection devices havegenerally been avoided because of the associated complexity,computational burden, and communications requirements. Where digitalcommunications can have different delays in different directions betweena pair of locations which lead to an error in clock synchronization.

Variable size data windows in power system protection devices havegenerally been avoided because of the associated complexity,computational burden, and communications requirements. Where variablesize data windows have been implemented, a different set of weightingfunctions is used for each data window. When the data window changessize, recalculations are then required for all the samples in the datawindow.

Conventional power system impedance relays, including electromechanical,solid state, and digital relays, typically detect faults by calculatingan effective impedance from voltage and current measurements. When theeffective impedance falls within a certain range, a fault is declared.For a first zone relay, the range is typically set for less than 85-90%of the total line length impedance to allow for uncertainties in theunderlying measurements of power system quantities. The actualuncertainties vary with time. Conventional impedance relays do notrecognize the time varying quality of the underlying measurements, andthus sensitivity and security can be compromised.

Inherent uncertainties exist in estimating fundamental power frequencyvoltages and currents from digitized samples and arise from a number ofsources, including, for example, power system noise, transients, sensorgain, phase and saturation error, and sampling clock error. Conventionalpractice is to account for these errors during system design byestimating the worst case and including enough margin to allow for theerrors. The conventional procedure does not take into account the timevarying nature of the errors. Other procedures to determine the sum ofthe squares are computationally intensive.

The standard method for providing transformer current differentialprotection is to develop restraint and operate signals from measuredtransformer currents from each winding, and to use a discrete Fouriertransform (DFT) or a fast Fourier transform (FFT) to calculate variousharmonics. The operate signal is usually calculated based on theprinciple that the sum of the ampere turns approximately equals themagnetizing current and is thus calculated as the algebraic sum of theampere turns for each winding. The restraint signal is usually based onthe fundamental frequency current or a weighted sum of the fundamentalfrequency current and selected harmonics to factor magnetizing inrushand overexcitation.

SUMMARY

It would be desirable to have a digital differential current systemcapable of operating for a wide range of bandwidth communicationchannels with faster response time and increased sensitivity overconventional systems.

It would also be desirable to have methods for synchronizing powersystem measurements at multiple locations; for calculating thefundamental power system frequency component of voltages and currentsfrom digital data samples over a variable size data window; forcalculating uncertainties from power system quantity measurements insuch a manner that a reach (the setting of a distance relay) iscontinuously adapted to the quality of the measurements; and fordetermining the uncertainty in fundamental power system frequencymeasurements of voltages and currents by estimating the errors on-linefrom available information in a way that tracks the time-varying natureof the errors.

In the present invention, current measurements are transmitted by a dataconsolidation of a partial sum of the terms used in a discrete Fouriertransform (DFT) and the required digital communications bandwidth isthereby reduced; an adaptive restraint region is automatically adjustedusing statistical principles to reflect the confidence in currentmeasurements during changing system conditions; and samplingsynchronization can be achieved by analyzing data in the measuredcurrents.

Data consolidation involves the extraction of appropriate parameters tobe transmitted from raw samples of transmission line phase currents.Data consolidation can be used to achieve a balance between transientresponse and bandwidth requirements. Consolidation is possible along twodimensions: time and phase. Time consolidation combines a time sequenceof samples to reduce the required bandwidth. Phase consolidationcombines information from the three phases and the neutral. Phaseconsolidation is generally not used in digital systems wherein detectionof which phase is faulted is desired. Time consolidation reducescommunications bandwidth requirements and improves security byeliminating the possibility of falsely interpreting a single corrupteddata sample as a fault. The present invention includes a newconsolidation technique called “phaselets.” Phaselets are partial sumsof terms of a complete phasor calculation. Phaselets can be combinedinto phasors over any time window that is aligned with an integralnumber of phaselets. The number of phaselets that must be transmittedper cycle per phase is the number of samples per cycle divided by thenumber of samples per phaselet.

A restraint characteristic is the decision boundary between conditionsthat are declared to be a fault and those that are not. The presentinvention includes an adaptive decision process based on on-linecalculation of the sources of measurement error to create an ellipticalrestraint region having a variable major axis, minor axis, andorientation. Parameters of the ellipse vary with time to take advantageof the accuracy of the current measurements.

With respect to synchronization, the conventional technique, asdescribed in Mills, “Internet Time Synchronization: The Network TimeProtocol,” IEEE Transactions on Communications, vol. 39, no. 10, October1991, pages 1482-93, is a “ping-pong” technique which uses round triptime tag messages to synchronize clocks which calculate thecommunications delays. A limitation of the ping-pong technique is thatthe difference between the delays in each direction between twoterminals cannot be determined. The present invention includes a newtechnique for compensating for this uncertainty in the case of two orthree terminal transmission lines by using information in the measuredcurrents and digital communication. In this manner, measurement ofmagnitude and phase angle of power system voltages and currents atmultiple locations can be performed on a common time reference. Whenfour or more terminals are used, the conventional ping-pong technique isused.

BRIEF DESCRIPTION OF THE DRAWINGS

The features of the invention believed to be novel are set forth withparticularity in the appended claims. The invention itself, however,both as to organization and method of operation, together with furtherobjects and advantages thereof, may best be understood by reference tothe following description taken in conjunction with the accompanyingdrawings, where like numerals represent like components, in which:

FIG. 1 is a block diagram of one transmission line protection embodimentof the present invention.

FIG. 2 is a block diagram of another transmission line protectionembodiment of the present invention.

FIG. 3a is a circuit diagram of a two-terminal, single phase, equivalentline charging model.

FIG. 3b illustrates a three phase charge compensation model for aterminal.

FIG. 4 is a graph of a fitted sign wave and the sum of the squares ofthe errors between the measured data samples and the fitted sign wavewith respect to time.

FIG. 5 is a circuit diagram of a distance relay.

FIG. 6 is a block diagram of a transformer protection embodiment of thepresent invention.

DETAILED DESCRIPTION

Two types of architecture can be used in the present invention:master-remote and peer—peer. Furthermore, breaker-and-a-halfconfigurations can be used for either type of architecture, if desired.

In the master-remote embodiment 10, as shown in FIG. 1, a single masterdevice 12 (having a clock 12 a) at a terminal 30 maintainssynchronization of remote clocks 14 a, 16 a, and 18 a at remote devices14, 16, and 18 of terminals 24, 26, and 28, respectively, receivescurrent measurement from the remote devices as well as local current,and identifies fault conditions on power line 20. Remote devices measureterminal currents using current sensors 32, 34, 36, and 38 for eachphase, convert samples to phaselets, and communicate phaseletinformation and measurement uncertainty information with the masterdevice along communication lines 22 a, 22 b, 22 c, 22 d, 22 e, and 22 f.Preferably two communication lines are present between each remotedevice and the master device for communication redundancy purposes. Inaddition to a respective current sensor, each terminal 30, 24, 26, and28 also includes, among other components, a respective circuit breaker30 a, 24 a, 26 a, and 28 a and a respective bus 30 b, 24 b, 26 b, and 28b.

The master device can be physically situated anywhere in the powersystem. To minimize round trip communication delays, a preferredlocation is central to all ends of the transmission line. The master canbe located near one terminal, for example. A remote device is situatedat each terminal. In the case of a co-located master and remote, asshown, the functions can be combined into a single device 12.

In the peer—peer embodiment, as shown in FIG. 2, a plurality ofterminals 46, 48, and 50 (each including respective circuit breakers 46a, 48 a, and 50 a and respective buses 46 b, 48 b, and 50 b) have powerlines 58 monitored by current sensors 52, 54, and 56 of peers 40, 42,and 44, respectively. Each peer has communication lines (shown as 60 a,60 b, and 60 c) extending to at least some of the other peers and isadapted to perform current analysis in a manner similar to that of themaster embodiment discussed above. A single communication line betweeneach pair of peers is sufficient. Not every pair of peers requires acommunication line therebetween, especially in the case of four or moreterminals. The communication lines should be chosen so that the systemis operable even if one of the lines fails.

At each terminal of FIG. 1 or FIG. 2, the three phase currents aresampled a number (N) of times per cycle. If desired, ground current canbe derived from phase currents at the master or peer. Roughsynchronization can be maintained by use of the ping-pong messagingtechnique. For two and three terminal systems, more accuratesynchronization can be accomplished by examining the sum of the currentphasors.

Decaying offset can be removed from each phase measurement using adigital simulation of a circuit commonly referred to as a “mimiccircuit,” which is based on the differential equation of the inductivecircuit which generates the offset. Then phaselets are calculated ateach terminal for each phase of the current (or, if decaying offset hasbeen removed, from output signals of the mimic calculation), and the sumof the squares of the raw data samples is calculated for each phase.

Phaselets are combined into phasors, and ground current can bereconstructed from phase information if desired. An elliptical restraintregion is determined by combining sources of measurement error. When adisturbance is detected, the variable size calculation window isreadjusted to leave out pre-fault current measurements from the phasordetermination.

A fault is indicated by the detection of a disturbance and by the sum ofthe current phasors falling outside of the elliptical restraint region.The statistical distance from the phasor to the restraint region can bean indication of the severity of the fault. To provide speed of responsethat is commensurate with fault severity, the distance can be filteredusing a single pole low pass filter of about 60 hertz, for example. Formild faults, filtering improves measurement precision at the expense ofa slight delay on the order of one cycle. Severe faults can be detectedwithin a single phaselet.

Whenever the sum of phasors falls within the elliptical restraintregion, the system assumes there is no fault, and uses whateverinformation is available for fine adjustment of the clocks.

Time synchronization

In addition to being important for multi-terminal transmission lines,time synchronization is important in many other applications such aspower relays, determinations of sequences of events, economic powerdispatch, and any other situation requiring synchronization of clocks.The synchronization technique discussed herein can be applied tosynchronize clocks at the terminals of a two or a three terminal systemby examining the sum of the positive sequence currents measured at theterminals. In some situations, larger collections of clocks can then besynchronized by taking advantage of the fact that clocks at the samelocation can share data and be synchronized. Synchronization errors showup as phase angle and transient errors in phasor measurements at theterminal. Phase angle errors occur when identical currents producephasors with different phase angles, and transient errors occur whencurrents change at the same time and the effect is observed at differenttimes at different measurement points.

For systems having four more terminals as well as systems having two orthree terminals under conditions where no current is flowing, theconventional ping-pong technique will be used. The amount of timesynchronization error in the ping-pong procedure depends on factorsincluding the stability of the local clocks, how often the ping-pong isexecuted, and differential channel delay. The ping-pong must be executedoften enough to compensate for drifts in the local clocks. A smallamount of channel delay itself is not critical (mainly affecting onlysystem transient response) provided that the channel delay is the samein each direction between terminals. If the channel delay is not thesame, the difference between the delays causes a differential errorbetween the clocks being synchronized over the restraint boundary andreduces the system sensitivity. Thus, in the case of four or moreterminals, the differential delay should be specified and controlled toachieve design goals.

In the case of two or three terminals, additional information isextracted from the current phasors to determine phase angle errors. Thebasis for synchronizing clocks at the termination points of atransmission line is that, according to fundamental circuit theory laws,the sum of the positive sequence currents is equal to the positivesequence charging current for the transmission lines. The positivesequence charging current can be calculated from measured voltages.Inequalities are attributable to errors in the magnitudes and/or phaseangles of the estimates of the positive sequence currents. In the caseof a two or a three terminal transmission line, phase angle errors,which depend on synchronization errors, can be determined approximatelyfor each terminal.

Data sampling can additionally be synchronized to the power systemfrequency to eliminate the error effects of asynchronous sampling.Terminal clocks are phase locked with each other and frequency locked tothe power system. The basic method of frequency locking is to calculatefrequency deviation from the apparent rotation of phasors in the complexplane and adjust the sampling frequency accordingly. This calculationoccurs in the master terminal for a master-remote architecture and inall terminals that serve as time and frequency references for apeer-to-peer architecture. The rotational rate of phasors is equal tothe difference between the power system frequency and the ratio of thesampling frequency divided by the number of samples per cycle. Thisdifference is used to correct the sampling clocks to synchronize thesampling with the power system frequency. The correction is calculatedonce per power system cycle. For conciseness, a phasor notation is usedas follows: $\begin{matrix}{{{\overset{\_}{I}(n)} = {{PhasorReal}_{n} + {j \cdot {PhasorImaginary}_{n}}}},} & (1) \\{{{{\overset{\_}{I}}_{a,k}(n)} = {{\overset{\_}{I}(n)}\quad {for}\quad {phase}\quad a\quad {from}\quad {the}\quad {k{th}}\quad {terminal}\quad {at}\quad {time}}}\quad {{{step}\quad n},}} & (2) \\{{{{\overset{\_}{I}}_{b,k}(n)} = {{\overset{\_}{I}(n)}\quad {for}\quad {phase}\quad b\quad {from}\quad {the}\quad {k{th}}\quad {terminal}\quad {at}\quad {time}}}\quad {{{step}\quad n},}} & (3) \\{{{{\overset{\_}{I}}_{c,k}(n)} = {{\overset{\_}{I}(n)}\quad {for}\quad {phase}\quad c\quad {from}\quad {the}\quad {k{th}}\quad {terminal}\quad {at}\quad {time}}}\quad {{{step}\quad n},}} & (4)\end{matrix}$

The positive sequence current can then be calculated for each terminalby the following equation: $\begin{matrix}{{{{\overset{\_}{I}}_{{pos},k}(n)} = {{1/3} \cdot ( {{{\overset{\_}{I}}_{a,k}(n)} + {^{\frac{j\quad 2\pi}{e}}{{\overset{\_}{I}}_{b,k}(n)}} + {^{- \frac{j\quad 2\pi}{3}}{{\overset{\_}{I}}_{c,k}(n)}}} )}},} & (5)\end{matrix}$

wherein n is the sample number at the kth terminal of the transmissionline.

The contribution of charging currents can be removed at each respectiveterminal by subtraction. FIG. 3a illustrates a two-terminal, positivesequence, equivalent line charging model, and FIG. 3b illustrates thethree phase charge compensation model for a terminal.

In the case of a power system transmission line having line resistance66 and inductance 68, the sum of the currents entering terminals 70 and72 is not exactly zero because of the capacitive charging current forthe line. For short transmission lines, the charging current can betreated as an unknown error. In these embodiments, no voltage sensorsare needed, and line charging current is included as a constant term inthe total variance (discussed with respect to equation 37 below),thereby increasing the restraint of the system to compensate for theline charging current.

For long transmission lines, the charging current can become significantand compromise the sensitivity of the differential algorithm, socharging current compensation using voltage measurements is beneficial.One technique for such compensation is to subtract a C dv/dt term(capacitance 62 or 64 multiplied by the change in voltage over time)from the measured current at each terminal of the system. This techniqueprovides compensation of the capacitive current at both the fundamentalpower system frequency and some of the frequencies of the transientresponse of the transmission line. The fine details of traveling waveson the transmission line are not compensated for, and they contribute tothe restraint by increasing the sum of the squares of the errors in thedata samples. Although a model for compensation for a two terminalsystem is shown, the model can be extended to accommodate any number ofterminals.

When three phase models are used, as shown in FIG. 3b, both phase tophase capacitance (Cpp) and phase to ground capacitance (Cpg) must beanalyzed. In terms of zero sequence and positive sequence capacitance,Cpg and Cpp are given by Cpg=Czero (zero sequence capacitance) and Cpp=⅓Cplus (positive sequence capacitance) minus ⅓ Czero. The compensationtechnique for each phase can use data from all three phases. Forexample, the compensation for phase “a” can be provided byCpg*dVa/dt+Cpp*(2*dVa/dt−dVb/dt−dVc/dt), wherein Va, Vb, and Vc arephase voltages. Another equivalent expression for the phase “a” chargingcurrent is Cplus*(dVa/dt−dVo/dt)+Czero*dVo/dt, wherein Vo is the zerosequence voltage.

For some very long lines, the distributed nature of the lines lead tothe classical transmission line equations, which can be solved forvoltage and current profiles along the line. The compensation model usesthe effective positive and zero sequence capacitance seen at theterminals of the line.

In some applications with long transmission lines, shunt reactors can beused to provide some of the charging current required by the line. Theshunt reactors reduce the amount of charging current seen by thedifferential system at the fundamental power system frequency.Additionally, the shunt reactors interact with the charging capacitanceto introduce additional frequency components in the transient responseof the transmission line. In one embodiment, the protection chargingcompensation is set to equal the residual charging current (thedifference of capacitive and inductive reactance) at the fundamentalpower system frequency. The inductor current can be effectively“removed” from the circuit via a current transformer connection (notshown).

The basic procedure for achieving improved phase synchronism is to makesmall adjustments in the sampling clocks to drive the sum of currentphasors toward zero. Because synchronization error affects all threephases, these adjustments can be based on positive sequence currents.

In the case of a two terminal system, the clock phase angle corrections(φ₁(n),φ₂(n)) are calculated from the positive sequence currents asfollows: $\begin{matrix}{{{\varphi_{1}(n)} = {{\frac{1}{2} \cdot \arctan}\quad ( \frac{{imag}\quad ( {{{\overset{\_}{I}}_{{pos},2}(n)} \cdot {{\overset{\_}{I}}_{{pos},1}^{*}(n)}} )}{{real}( {{{\overset{\_}{I}}_{{pos},2}(n)} \cdot {{\overset{\_}{I}}_{{pos},1}^{*}(n)}} )} )}},\quad {and}} & (6) \\{{\varphi_{2}(n)} = {- {{\varphi_{1}(n)}.}}} & (7)\end{matrix}$

It is possible to use a four quadrant arc tangent, in which case theminus signs are needed on the imaginary and real part as shown.

In the case of a three terminal system, the corrections(φ₁(n),φ₂(n),φ₃(n)) are approximated by the following equations:$\begin{matrix}{{{\varphi_{1}(n)} \approx \frac{\begin{matrix}{{real}( {( {{{\overset{\_}{I}}_{{pos},2}(n)} - {{\overset{\_}{I}}_{{pos},3}(n)}} ) \cdot} } \\ ( {{{\overset{\_}{I}}_{{pos},1}^{*}(n)} + {{\overset{\_}{I}}_{{pos},2}^{*}(n)} + {{\overset{\_}{I}}_{{pos},3}^{*}(n)}} ) )\end{matrix}}{\begin{matrix}{{imag}( {{{{\overset{\_}{I}}_{{pos},2}(n)} \cdot {{\overset{\_}{I}}_{{pos},1}^{*}(n)}} + {{{\overset{\_}{I}}_{{pos},3}(n)} \cdot}} } \\ {{{\overset{\_}{I}}_{{pos},2}^{*}(n)} + {{{\overset{\_}{I}}_{{pos},1}(n)} \cdot {{\overset{\_}{I}}_{{pos},3}^{*}(n)}}} )\end{matrix}}},} & (8) \\{{{\varphi_{2}(n)} \approx \frac{\begin{matrix}{{real}( {( {{{\overset{\_}{I}}_{{pos},3}(n)} - {{\overset{\_}{I}}_{{pos},1}(n)}} ) \cdot} } \\ ( {{{\overset{\_}{I}}_{{pos},1}^{*}(n)} + {{\overset{\_}{I}}_{{pos},2}^{*}(n)} + {{\overset{\_}{I}}_{{pos},3}^{*}(n)}} ) )\end{matrix}}{\begin{matrix}{{imag}( {{{{\overset{\_}{I}}_{{pos},2}(n)} \cdot {{\overset{\_}{I}}_{{pos},1}^{*}(n)}} + {{{\overset{\_}{I}}_{{pos},3}(n)} \cdot}} } \\ {{{\overset{\_}{I}}_{{pos},2}^{*}(n)} + {{{\overset{\_}{I}}_{{pos},1}(n)} \cdot {{\overset{\_}{I}}_{{pos},3}^{*}(n)}}} )\end{matrix}}},} & (9) \\{{\varphi_{3}(n)} \approx {\frac{\begin{matrix}{{real}( {( {{{\overset{\_}{I}}_{{pos},1}(n)} - {{\overset{\_}{I}}_{{pos},2}(n)}} ) \cdot} } \\ ( {{{\overset{\_}{I}}_{{pos},1}^{*}(n)} + {{\overset{\_}{I}}_{{pos},2}^{*}(n)} + {{\overset{\_}{I}}_{{pos},3}^{*}(n)}} ) )\end{matrix}}{\begin{matrix}{{imag}( {{{{\overset{\_}{I}}_{{pos},2}(n)} \cdot {{\overset{\_}{I}}_{{pos},1}^{*}(n)}} + {{{\overset{\_}{I}}_{{pos},3}(n)} \cdot}} } \\ {{{\overset{\_}{I}}_{{pos},2}^{*}(n)} + {{{\overset{\_}{I}}_{{pos},1}(n)} \cdot {{\overset{\_}{I}}_{{pos},3}^{*}(n)}}} )\end{matrix}}.}} & (10)\end{matrix}$

The phase angle corrections are filtered to slowly adjust the clocks ateach terminal until the phase angle corrections are driven toward zeroin a closed loop technique. This adjustment of the phase angles canoccur either by itself or in combination with the frequencysynchronization technique discussed below.

For each terminal, a quantity can be derived from the positive sequencecurrent (with or without removal of the charging current—depending onthe application) that is indicative of the amount of rotation from onecycle to the next by calculating the product of the positive sequencecurrent and the complex conjugate of the positive sequence current fromthe previous cycle:

Deviation={overscore (I)} _(pos.k)(n)·({overscore (I)}_(pos.k)(n−N))*  (11).

The angle of the deviation phasor per cycle for each terminal isproportional to the frequency deviation at that terminal, as discussedin commonly assigned Premerlani, U.S. Pat. No. 4,715,000, issued Dec.22, 1987. Because the clock synchronization method maintains frequencysynchronism, the frequency deviation is the same for each terminal.Therefore, the deviation phasors from all of the terminals can be addedto obtain a net deviation phasor for the system: $\begin{matrix}{{{{Deviation}(n)} = {\sum\limits_{k = 1}^{NT}{{Deviation}_{k}(n)}}},} & (12)\end{matrix}$

wherein NT is the number of terminals. The deviation phasors arefiltered to reduce the effects of noise and control the transientbehavior of the adaptation process. Then the sampling deviationfrequency is calculated from the filtered deviation phasorDeviationf(n): $\begin{matrix}{\frac{\Delta \quad f}{f_{o}} = {\arctan \quad ( \frac{{imag}( {{Deviationf}(n)} }{{real}( {{Deviationf}(n)} } )}} & (13)\end{matrix}$

wherein Δf is the frequency deviation and f_(o) is the nominalfrequency. A four quadrant arc tangent can be calculated by taking theimaginary and the real part of the deviation separately for the twoarguments of the four quadrant arc tangent. The sampling clocks of theterminals can be adjusted to drive the sampling frequency deviationtoward zero.

Data Consolidation

The fundamental power system frequency voltages and currents arecalculated from digitized samples of voltages and currents with aminimized computational burden and a data window which can have variablesizes. In applications requiring the communication of voltage andcurrent information, the method is particularly efficient with respectto communication bandwidth requirements. This invention can be used indigital devices that measure fundamental frequency voltage or currentcomponents.

The data compression minimizes the sum of the squares of the errorsbetween data samples and a sinusoidal waveform that best approximatesthe data samples. In the case of a fixed size data window that is anintegral multiple of a half cycle, a discrete Fourier transform (DFT)which can be modified for use according to the present invention. For avariable size data window, a modified DFT presents problems such as thecalculation of a complex amplitude from a weighted sum of the datasamples when the weights of individual samples depend on the width ofthe data window, as well as the accompanying bandwidth requirements.

The “phaselet” technique of the present invention partitions thecalculation into two processes. The first process is a calculation ofpartial sums of the data samples multiplied by one cycle weights. Thesecond process is a summation of the partial sums over the width of thedesired data window and a correction for distortion caused by the onecycle weights.

The partial sums (phaselets) are calculated by multiplying each datasample by a corresponding complex factor that does not depend on thedata window, and then adding the results over several data cycles. Inapplications requiring communications of complex amplitudes, phaseletsare communicated as they are calculated. The number of samples in aphaselet can be selected to achieve whatever compromise is desiredbetween communications bandwidth requirements and transient response. Alarge number of samples per phaselet reduces communications bandwidth. Asmall number of samples per phaselet reduces transient response.

Phaselets are converted into complex amplitudes over any size datawindow by adding the phaselets in that window and multiplying byprecalculated factors for that particular window.

More specifically, a phaselet is a portion of the sine and cosineweighted sum of data samples. Over a half cycle, a phasor is calculatedfrom phaselets by simply adding phaselets and multiplying by 4/N. Forwindow sizes other than a half cycle or multiple of a half cycle, aphasor is calculated by adding phaselets over the window and thenmultiplying a normalization two-by-two real matrix by the real andimaginary portions of the sum of phaselets.

Phasors are represented as real and imaginary components. The realcomponent represents the cos(ωt) term, and the imaginary componentsrepresent minus the sin(ωt) term. The convention used herein is torepresent all phasor quantities as peak values. To convert to rms,divide by the square root of two.

In the following equations, all indexes in summations increment by 1from the lower limit of the summation up to an including the upperlimit. Counting of samples, phaselets, and cycles starts from 1, whichis assumed to be the time reference t=0. The arguments of the cosine andsine functions are offset by ½ the sampling angle. The firstcoefficient, for time t=0, integer index (k) count=1, is for a phaseangle of ½ the sampling angle. Sampling occurs at the rate of N samplesper cycle, where the choice of N depends on the particular application.Phaselets are calculated every P data samples, where the choice of Pdepends on the application requirements. Phasors are updated whenevernew phaselets are available. In the case of a ½ cycle sliding window, afixed number of phaselets are added to produce a phasor. When a variablewindow is used, the number of phaselets is proportional to the windowsize.

A phaselet is a sum of sine and cosine weighted samples over a fractionof a cycle. An appropriate way to extract 16 phasor values per cyclefrom 64 samples per cycle is to first calculate sine and cosine weightedsums of groups of 4 samples per group. A DFT over a ½ cycle could becalculated by adding phaselets over the ½ cycle and multiplying by asuitable factor. For other size windows, a DFT is calculated by addingphasors and multiplying by an appropriate matrix that depends on thewidth and phase angle of the window.

Phaselets are calculated as follows: $\begin{matrix}{{{PhaseletReal}_{p} = {\sum\limits_{k = {{p \cdot P} - P + 1}}^{p \cdot P}{\cos \quad {( {\frac{2 \cdot \pi}{N} \cdot ( {k - \frac{1}{2}} )} ) \cdot x_{k}}}}},{and}} & (14) \\{{{PhaseletImaginary}_{p} = {- {\sum\limits_{k = {{p \cdot P} - P + 1}}^{p \cdot P}{\sin \quad {( {\frac{2 \cdot \pi}{N} \cdot ( {k - \frac{1}{2}} )} ) \cdot x_{k}}}}}},} & (15)\end{matrix}$

wherein

PhaseletReal_(p)=the real part of the pth phaselet for signal x,

PhaseletImaginary_(p)=the imaginary part of the pth phaselet for x,

p=phaselet index; there are N/P phaselets per cycle,

P=number of samples per phaselet,

N=number of samples per cycle,

X_(k)=kth sample of signal x, taken at N samples per cycle.

The argument to the cosine and sine function in equations 14 and 15 isoffset by ½ the sampling angle for simplifying the calculation of thematrices needed to convert phaselets into phasors and simplifying anoptional process for correcting for a small error in the samplingfrequency.

Various numbers of phaselets can be combined to form a sliding windowDFT. For a ½ cycle DFT, for example, the process for calculating phasorsfrom phaselets is adding phaselets and multiplying by 4/N:$\begin{matrix}{{{PhasorReal}_{n} = {\frac{4}{N} \cdot ( {\sum\limits_{p = {n - \frac{N}{2 \cdot P} + 1}}^{n}{PhaseletReal}_{p}} )}},{and}} & (16) \\{{{PhasorImaginary}_{n} = {\frac{4}{N} \cdot ( {\sum\limits_{p = {n - \frac{N}{2 \cdot P} + 1}}^{n}{PhaseletImaginary}_{p}} )}},} & (17)\end{matrix}$

wherein n represents the phasor index (there are N/P phasors per cycle).

It is possible to perform the sums recursively in fixed point arithmeticwithout fear of accumulated round off errors, provided that themultiplications are performed on a sample by sample basis. Afterinitialization, the sums at one value of n are calculated from theprevious sums by adding the newest terms of the new sums and subtractingthe oldest terms of the old sums.

Converting phaselets into phasors can also be done for other windowsizes by adding phaselets and then multiplying by a phaselettransformation matrix. First the phaselets are added together over thedesired window: $\begin{matrix}{{{PhasorSumReal}_{n} = {\sum\limits_{p = {n - \frac{W}{P} + 1}}^{n}{PhaseletReal}_{p}}},{and}} & (18) \\{{{PhasosSumImaginary}_{n} = {\sum\limits_{p = {n - \frac{W}{P} + 1}}^{n}{PhaseletImaginary}_{p}}},} & (19)\end{matrix}$

wherein W=window size in samples, and W/P is window size in phaselets.

Optionally, these sums can be calculated recursively. As discussedabove, the sums at one value of n can be calculated from the previoussums by adding the newest terms of the new sums and subtracting theoldest terms of the old sums. Sums are then converted into phasors bymultiplying by the following matrix: $\begin{matrix}\begin{matrix}{\begin{bmatrix}{PhasorReal}_{n} \\{PhasorImaginary}_{n}\end{bmatrix} = \quad {\begin{bmatrix}{T_{RR}( {n,W} )} & {T_{RI}( {n,W} )} \\{T_{IR}( {n,W} )} & {T_{II}( {n,W} )}\end{bmatrix} \cdot}} \\{\quad \begin{bmatrix}{PhaseletSumReal}_{n} \\{PhaseletSumImaginary}_{n}\end{bmatrix}}\end{matrix} & (20) \\{wherein} & \quad \\\begin{matrix}{T_{RR} = \quad {( {1 - ( {{\frac{1}{W} \cdot \cos}\quad {( {\frac{4 \cdot \pi \cdot P \cdot n}{N} - \frac{2 \cdot \pi \cdot W}{N}} ) \cdot \frac{\sin ( \frac{2 \cdot \pi \cdot W}{N} )}{\sin ( \frac{2 \cdot \pi}{N} )}}} )} ) \cdot}} \\{\quad \frac{2}{W - \frac{( {\sin ( \frac{2 \cdot \pi \cdot W}{N} )} )^{2}}{W \cdot ( {\sin ( \frac{2 \cdot \pi}{N} )} )^{2}}}}\end{matrix} & (21) \\\begin{matrix}{T_{RI} = \quad {T_{IR} = {{\frac{1}{W} \cdot \sin}\quad {( {\frac{4 \cdot \pi \cdot P \cdot n}{N} - \frac{2 \cdot \pi \cdot W}{N}} ) \cdot}}}} \\{\quad {\frac{\sin ( \frac{2 \cdot \pi \cdot W}{N} )}{\sin ( \frac{2 \cdot \pi}{N} )} \cdot \frac{2}{W - \frac{( {\sin ( \frac{2 \cdot \pi \cdot W}{N} )} )^{2}}{W \cdot ( {\sin ( \frac{2 \cdot \pi}{N} )} )^{2}}}}}\end{matrix} & (22) \\\begin{matrix}{T_{II} = \quad {( {1 + ( {{\frac{1}{W} \cdot \cos}\quad {( {\frac{4 \cdot \pi \cdot P \cdot n}{N} - \frac{2 \cdot \pi \cdot W}{N}} ) \cdot \frac{\sin ( \frac{2 \cdot \pi \cdot W}{N} )}{\sin ( \frac{2 \cdot \pi}{N} )}}} )} ) \cdot}} \\{\quad {\frac{2}{W - \frac{( {\sin ( \frac{2 \cdot \pi \cdot W}{N} )} )^{2}}{W \cdot ( {\sin ( \frac{2 \cdot \pi}{N} )} )^{2}}}.}}\end{matrix} & (23)\end{matrix}$

The matrix thus depends on design constants P and N and variables W andn. In principle, a matrix should be calculated for each combination of nand W.

Until a disturbance is detected, phaselets will be combined to form asliding window D. After a disturbance is detected, the window isre-initialized by removing the previously calculated phaselets from thecurrent window and then expanding the current window as new data isgathered to reform a sliding window DFT. In this manner, the morerelevant disturbance information is not diluted by the presence ofearlier obtained normal information.

Removal of Decaying Offsets

In some applications decaying offset can occur and create errors whichinterfere with the determination of how well the measured currentsamples fit a sine wave. For example, the inductive behavior of powersystem transmission lines gives rise to decaying exponential offsetsduring transient conditions.

A mimic algorithm can be used to remove decaying offsets as follows:$\begin{matrix}{{{Imimic}_{m} = {{\frac{R}{\cos ( \frac{\pi \cdot M}{N} )} \cdot \frac{( {i_{m} + i_{m - M}} )}{2}} + {\frac{X}{\sin ( \frac{\pi \cdot M}{N} )} \cdot \frac{( {i_{m} - i_{m - M}} )}{2}}}},} & (24)\end{matrix}$

wherein

Imimic_(m)=mth sample of the output of the mimic algorithm,

i_(m)=mth current sample,

m=sample index, starting from 1, at N samples per cycle,

M=interval, in samples, used to approximate the mimic simulation,

N=sampling rate, samples per cycle,

X=reactance of the mimic,

R=resistance of the mimic.

A similar equation can be used for voltage samples. Transient responseof the mimic is M samples, at N samples per cycle. Using M=4 and N=64,transient response is 1 millisecond.

Next the phaselets are calculated in the same manner as equations 14 and15 with the substitution of Imimic_(k) for x_(k) as follows:$\begin{matrix}{{{PhaseletReal}_{p} = {\sum\limits_{k = {{p \cdot P} - P + 1}}^{p \cdot P}{\cos \quad {( {\frac{2 \cdot \pi}{N} \cdot ( {k - \frac{1}{2}} )} ) \cdot {Imimic}_{k}}}}},{and}} & (25) \\{{{PhaseletImaginary}_{p} = {\sum\limits_{k = {{p \cdot P} - P + 1}}^{p \cdot P}{{- \sin}\quad {( {\frac{2 \cdot \pi}{N} \cdot ( {k - \frac{1}{2}} )} ) \cdot {Imimic}_{k}}}}},} & (26)\end{matrix}$

wherein Imimic_(m)=mth sample of the output of the mimic algorithm.

In order to calculate statistical parameters, the sum of the squares ofindividual samples is also calculated: $\begin{matrix}{{PartialSumOfSquares}_{p} = {\sum\limits_{k = {{p \cdot P} - P + 1}}^{p \cdot P}{{Imimic}_{k}^{2}.}}} & (27)\end{matrix}$

As discussed above, until a disturbance is detected, phaselets will becombined to form a sliding window DFT, and after a disturbance isdetected, the window is re-initialized by removing the previouslycalculated phaselets from the current window and then expanding thecurrent window as new data is gathered to reform a sliding window DFT.

A sliding window DFT can be of a one half cycle as discussed above or amultiple of one half as discussed here with respect to the one cyclesliding window. A one half cycle window provides a faster transientresponse but does not have as high an accuracy as a one cycle window.

For a one cycle DFT, the process for calculating phasors from phaseletsand the sum of squares form partial sums is simple, as shown in thefollowing equations: $\begin{matrix}{{{PhasorReal}_{n} = {\frac{2}{N} \cdot ( {\sum\limits_{p = {n - \frac{N}{P} + 1}}^{p \cdot P}{PhaseletReal}_{p}} )}},{and}} & (28) \\{{{PhasorImaginary}_{n} = {\frac{2}{N} \cdot ( {\sum\limits_{p = {n - \frac{N}{P} + 1}}^{p \cdot P}{PhaseletImaginary}_{p}} )}},} & (29)\end{matrix}$

wherein

PhasorReal_(n)=the real part of the nth phasor,

PhasorImaginary_(n)=the imaginary part of the nth phasor,

n=phasor index; there are N/P phasors per cycle.

In order to calculate statistical parameters, the sum of the squares ofindividual samples is also calculated: $\begin{matrix}{{{SumOfSquares}_{n} = {\sum\limits_{p = {n - \frac{N}{P} + 1}}^{n}{{Partial}\quad {SumOfSquares}_{p}}}},} & (30)\end{matrix}$

wherein SumOfSquares_(n)=the nth sum of squares.

The above equations are defining. The sums involved are not actuallycalculated in the order shown, but are calculated recursively. Afterinitialization, the sums at one value of n are calculated from theprevious sums by adding the newest terms of the new sums and subtractingthe oldest terms of the old sums which would fall outside the currentwindow.

As discussed above, converting phaselets to phasors can also be done forother window sizes by adding phaselets and then multiplying by anormalization matrix. First the phaselets are added together over thedesired window using equations 18 and 19 to obtain PhaseletSumReal_(n)and PhaseletSumImaginary_(n). Then the sum of the squares is calculatedas follows: $\begin{matrix}{{SumOfSquares}_{n} = {\sum\limits_{p = {n - \frac{W}{P} + 1}}^{n}{{Partial}\quad {{SumOfSquares}_{p}.}}}} & (31)\end{matrix}$

Phaselets sums are converted into phasors by multiplying by theprecalculated matrix discussed with respect to equations 20 through 23.

Sum of the squares of errors between measured data samples and thefitted sine wave

Although the sum of the squares of errors is discussed herein in thecontext of multi-terminal transmission lines, other applications inwhich the invention an be useful include, for example, motor protection,control and diagnostic devices such as turn fault detectors, powersystem relays such as distance, transformer, bus, and generator relays,industrial protective devices, and drive systems.

The sum of squares of the errors between measured data samples and thefitted sine wave can be calculated from the sum of squares, thephaselets, and the phasors as follows: $\begin{matrix}{E_{n}^{2} = \quad {{SumOfSquares}_{n} - {\begin{pmatrix}{{{PhaseletSumReal}_{n} \cdot {PhasorReal}_{n}} +} \\{{PhaseletSumImaginary}_{n} \cdot {PhasorImaginary}_{n}}\end{pmatrix}.}}} & (32)\end{matrix}$

This method for calculating the sum of the squares of the errors isvalid for any size data window. The methods used in the prior art eitherrequire more calculation or are valid only for data windows that areintegral numbers of a half cycle.

The equation for the squared error E_(n) ² is particularly efficientbecause it requires only two multiplies and three additions. Whenphaselets are already being calculated for data consolidation purposes,the only extra calculations will be the sum of the squares. All threesums (of phaselet real, phaselet imaginary, and squares) can becalculated recursively for a sliding window, even as the window changessize. Once a sum for a value of n is calculated, the value of the sumfor the next value of n can be calculated by adding the newest terms andsubtracting the oldest terms. For the equation to be applied properly infixed point arithmetic, enough bits must be used to hold squared values,and care must be taken with the scaling. Both the sum of the squares andthe phaselet sums are proportional to the number of samples in thewindow.

FIG. 4 is a graph illustrating a fitted sign wave representative of thephasors, and the sum of squares of the errors (E_(n) ²), with respect totime. A one cycle sliding window is represented at different points intime by T_(a), T_(b), and T_(c). At point T_(s) ¹, a disturbance isdetected and is reflected by a spike in the sum of squares of theerrors. When such disturbance condition is detected, the sliding windowis re-initialized. A new window, T_(s) ¹ (including only one phaselet),begins and does not include any previous phaselet information. As newsamples and phaselets are calculated, the new window expands. Forexample, after the next phaselet is calculated the new window, T_(s) ²,includes two phaselets. The window continues to expand with newlycalculated phaselets until a one cycle window is formed. At this pointthe window begins to slide again by adding the new terms and droppingthe oldest terms.

The sum of the squares of the errors is a sensitive disturbancedetection indicator. Other potential disturbance conditions such asmagnitude above a threshold of positive sequence current, negativesequence current, or ground current or changes in positive, negative, orzero sequence current or load current can be monitored and compared torespective thresholds to detect disturbances, if desired, in addition tocalculating the sum of the squares of the errors.

By itself, the two-norm equation 32 is only an indication of the sum ofthe squares of the sample errors. A more useful quantity is an estimateof the standard deviation of the samples: $\begin{matrix}{{\sigma_{n} = \sqrt{\frac{E_{n}^{2}}{W}}},} & (33)\end{matrix}$

wherein W is the window size in samples. The variance matrix for avariable window is given by: $\begin{matrix}\begin{matrix}{\begin{bmatrix}C_{{RR}{({n,W})}} & C_{{RI}{({n,W})}} \\C_{{IR}{({n,W})}} & C_{{II}{({n,W})}}\end{bmatrix} = {\sigma_{n}^{2} \cdot ( \begin{bmatrix}T_{{RR}{({n,W})}} & T_{{RI}{({n,W})}} \\T_{{IR}{({n,W})}} & T_{{II}{({n,W})}}\end{bmatrix} )}} \\{= {\frac{E_{n}^{2}}{W} \cdot ( \begin{bmatrix}T_{{RR}{({n,W})}} & T_{{RI}{({n,W})}} \\T_{{IR}{({n,W})}} & T_{{II}{({n,W})}}\end{bmatrix} )}}\end{matrix} & (34)\end{matrix}$

wherein

T_(RR(n,W)), T_(RI(n,W)), T_(IR(n,W)), & T_(II(n,W)) are defined byequations 21-23,

C_(RR)=expected value of square of error in real part of a phasor,

C_(RI)=C_(IR)=expected value of product of errors in the real &imaginary, parts,

C_(II)=expected value of square of error in imaginary part of a phasor.

The variance matrix in a phasor measurement is the square of the sampledeviations times the transformation matrix given previously. This matrixdescribes an elliptical uncertainty region in general, although forwindows that are integer multiples of a half cycle, the off diagonalterms are zero and the real and imaginary elements are the same.Strictly speaking, the uncertainty of a phasor measurement should beevaluated in light of the full matrix. However, a worst case conditioncan be considered to simplify the calculations. If a phasor happens tohave a particular phase angle, it will maximize the uncertainty in itsmagnitude.

The above description relates to the communication and analysis of threephase data. If less baud is available as would be the situation in a9600 baud system, for example, then a positive sequence current phasorand variance matrix information of each respective terminal can betransmitted once per cycle for all three phases. The magnitude of thefactor multiplying each phase current in the computation of positivesequence current is ⅓, so the net variance in the positive sequencecurrent is {fraction (1/9)} times the sum of the variance parameters foreach phase.

Although the positive sequence current is the preferred parameter fortransmission, either the zero sequence current or the negative sequencecurrent could alternatively be used. The zero (I_(O)), positive (I₊),and negative (I⁻) sequence currents can be calculated as follows.

I _(O)=(⅓)*(I _(A) +I _(B) +I _(C))  (34a)

I ₊=(⅓)*(I _(A) +e ^(j2/3) I _(B) +e ^(j4/3) I _(C))  (34b)

I ⁻=(⅓)*(I _(A) +e ^(j4/3) I _(B) +e ^(j2/3) I _(C))  (34c),

wherein I_(A), I_(B), and I_(C) represent current phasors obtained fromthe measured phase currents.

A positive sequence sum of the squares E₊ can be calculated from thesums of the squares (E_(A), E_(B), E_(C)) of the phases as follows:

E ₊={fraction (1/9)}(E _(A) ² +E _(B) ² +E _(C) ²)  (34d).

The terms in the variance matrix (C_(RR+), C_(RI+), C_(IR+), C_(II+))can thus be calculated from the variance matrix terms for the individualphases A, B, and C:

C _(RR+)={fraction (1/9)}(C _(RR)(n,W,A)+C _(RR)(n,W,A)+C_(RR)(n,W,A))  (34e),

C _(RI+) C _(IR+)={fraction (1/9)}(C _(RI)(n,W,A)+C _(RI)(n,W,A)+C_(RI)(n,W,A))  (34f),

C _(II+)={fraction (1/9)}(C _(II)(n,W,A)+C _(II)(n,W,A)+C_(II)(n,W,A))  (34g).

The positive sequence current and the three variance parameters can betransmitted from each terminal for analysis.

Distance Relay Reach

FIG. 5 is a circuit diagram of a distance relay including terminals 76and 82, current sensors 74 and 84, voltage sensors 78 and 86, andprocessors 80 and 88. Conventional distance relays operate by measuringvoltage and current at one end of a line to calculate effectiveimpedance, which is proportional to line length. Conventional distancerelay applications and techniques are discussed, for example, in Phadkeand Thorp, Computer Relaying for Power Systems (Research Studies PressLTD. and John Wiley & Sons Inc. 1988).

Line length impedance can be determined under short circuit conditions.The effective impedance is used to determine a fault location by settinga reach (a percentage of the line length impedance), comparing theeffective impedance to the reach, and declaring a fault condition if theeffective impedance is less than the reach. Conventional power systemimpedance relays use a reach, for a first zone relay, which is typicallyset for less than 80-90% of the total line length impedance to allow foruncertainties in the underlying measurements of power system quantities.

The actual uncertainties vary with time, however. Because conventionalimpedance relays do not recognize the time varying quality of theunderlying measurements, sensitivity and security can be compromised.For example, during transient start up periods of the relay, a low reachis appropriate, and during period with low uncertainties, a reach higherthan 90% can be desirable.

In the present invention, for a particular standard deviation σ_(n), asdetermined in equation 33, a deviation Δ_(n) (uncertainty) in a nominalreach is given by: $\begin{matrix}{\Delta_{n} = {\sigma_{n} \cdot {\sqrt{\frac{2}{W - \frac{\sin ( \frac{2 \cdot \pi \cdot W}{N} )}{\sin ( \frac{2 \cdot \pi}{N} )}}}.}}} & (35)\end{matrix}$

The deviation can be normalized by the following equation:$\begin{matrix}\begin{matrix}{\Delta_{normalized} = {\sqrt{\frac{\sigma_{V}^{2}}{V^{2}} + \frac{\sigma_{I}^{2}}{I^{2}}} \cdot \sqrt{\frac{2}{W - \frac{\sin ( \frac{2 \cdot \pi \cdot W}{N} )}{\sin ( \frac{2 \cdot \pi}{N} )}}}}} \\{{= {\sqrt{\frac{E_{V}^{2}}{W \cdot V^{2}} + \frac{E_{I}^{2}}{W \cdot I^{2}}} \cdot \sqrt{\frac{2}{W - \frac{\sin ( \frac{2 \cdot \pi \cdot W}{N} )}{\sin ( \frac{2 \cdot \pi}{N} )}}}}},}\end{matrix} & \text{(35a)}\end{matrix}$

wherein Δ_(normalized) is the normalized deviation, σ_(v) is thestandard deviation of voltage V, σ₁ is the standard deviation of currentI, E_(v) is the sum of squares of errors between phase voltage samplesand a fitted sign wave representative of corresponding real andimaginary phasors, and E_(I) is the sum of squares of errors betweenphase current samples and a fitted sign wave representative ofcorresponding real and imaginary phasors.

The normalized deviation can be multiplied by a factor related to theconfidence interval (such as the number of standard deviations) and theassumed distribution of the error to obtain an uncertainty percentage.One example factor is four (4). The uncertainty percentage can besubtracted from one (1) with the result being multiplied by the nominalreach to obtain an adjusted reach. The adjusted reach can be multipliedby the predetermined line impedance to obtain an adjusted impedancevalue for comparison with the effective impedance.

Fault severity of differential system

Normally the sum of the current phasors from all terminals is zero foreach phase. A fault is detected for a phase when the sum of the currentphasors from each terminal for that phase falls outside of a dynamicelliptical restraint boundary for that phase, based on statisticalanalysis. The severity of the fault is calculated from the covarianceparameters and the sum of the current phasor for each phase as follows:$\begin{matrix}{{{Severity} = {{{PhasorReal}^{2} \cdot \sqrt{\frac{C_{II}}{C_{RR}}}} - {{PhasorReal} \cdot {PhasorImaginary} \cdot 2 \cdot \frac{C_{RI}}{\sqrt{C_{RR} \cdot C_{II}}}} + {{PhasorImaginary}^{2} \cdot \sqrt{\frac{C_{RR}}{C_{II}}}} - {18 \cdot {Restraint}^{2} \cdot \sqrt{C_{RR} \cdot C_{II}} \cdot ( {1 - \frac{C_{RI}^{2}}{C_{RR} \cdot C_{II}}} )}}},} & (36)\end{matrix}$

wherein Restraint is a restraint multiplier analogous to the slopesetting of traditional differential techniques for adjusting thesensitivity of the relay. A value of 1 is recommended for this parameterfor most applications.

Raising the restraint multiplier corresponds statistically to demandinga greater confidence interval and has the effect of decreasingsensitivity. Lowering the restraint multiplier is equivalent to relaxingthe confidence interval and increases sensitivity. Thus, the restraintmultiplier is an application adjustment that is used to achieve thedesired balance between sensitivity and security.

Equation 36 is based on the covariance matrix and provides an ellipticalrestraint characteristic. When the covariance of the currentmeasurements is small, the restraint region shrinks. When the covarianceincreases, the restraint region grows to reflect the uncertainty of themeasurement. The calculated severity increases with the probability thatthe sum of the measured currents indicates a fault.

The second term of the severity equation arises from the orientation ofthe ellipse. The equation provides an adaptive elliptical restraintcharacteristic with the size, shape, and orientation of the ellipseadapting to power system conditions. The calculated severity is zerowhen the operate phasor is on the elliptical boundary, is negativeinside the boundary, and is positive outside the boundary. Outside ofthe restraint boundary, the computed severity grows as a square of thefault current. The restraint area grows as the square of the error inthe measurements.

The severity equation can optionally be filtered using, for example, asingle pole low pass filter having a time constant of several cycles.Such a filter can improve accuracy for high impedance faults.

On-Line Error Estimation in Power System Measurements

Although on-line error estimation is discussed herein in the context ofmulti-terminal transmission lines, other applications in which theinvention can be useful include, for example, motor protection, controland diagnostic devices such as turn fault detectors, power system relayssuch as distance, transformer, bus, and generator relays, industrialprotective devices, and drive systems.

To determine the uncertainty in fundamental power system frequencymeasurements of voltages and currents, the errors on-line are estimatedfrom available information in a way that tracks the time varying natureof the errors. Furthermore, bad samples are rejected in the case of anoccasional fault in the digital sampling electronics. The technique iswidely applicable and can be used anywhere fundamental power systemmeasurements are made, including control, protection, and monitoringdevices such as relays, meters, drive systems, and circuit breakers, forexample.

The method characterizes the uncertainty in a phasor estimate of afundamental frequency voltage or current with a two variable Gaussianprobability distribution with a time varying covariance matrix. This isa good approximation to the net effect of various sources of error evenif individual sources are not, strictly speaking, Gaussian. Thecovariance matrix is calculated for each source of error. Then, the netcovariance matrix is calculated by adding the matrices for all sources.The net covariance matrix can be used to characterize the uncertainty inthe calculation of any parameter that is derived from voltages orcurrents.

Typical sources of error include power system noise, transients, linecharging current, current sensor gain, phase and saturation error, clockerror, and asynchronous sampling. In some situations, some types oferrors, such as errors in the phase angle response of sensors and errorsdue to asynchronous sampling, can be driven to zero by other means. Forerrors that cannot be controlled, a covariance matrix is calculated foreach source of error for each phase. A total covariance matrix iscalculated for each phase by adding the matrices form each source. Theinvention treats various sources of errors as follows.

The system calculates the covariance matrix for errors caused by powersystem noise, harmonics, and transients. These errors arise becausepower system currents are not always exactly sinusoidal. The intensityof these errors varies with time, growing during fault conditions,switching operations, or load variations, for example. The system treatsthese errors as a Gaussian distribution in the real and in the imaginarypart of each phasor, with a standard deviation that is estimated fromthe sum of the squares of the differences between the data samples andthe sine function that is used to fit them. This error has a spectrum offrequencies. Current transformer saturation is included with noise andtransient error.

For current differential analysis, the preferred method for calculatingthe covariance matrix for noise, harmonics, transients, and currenttransformer saturation is from phaselets. The sum of the squares of theerrors in the data samples is calculated from the sum of the squaresinformation, phaselets, and phasor for each phase for each terminal ateach time step n using equation 32. The covariance matrix is thencalculated as a function of the time index and window size usingequation 34.

This covariance matrix is calculated separately for each phase of eachterminal. Total covariance for a phase due to this source of error isthe sum of the covariance matrices from each terminal for that phase.

Another source of error is a 60 Hz component of error current associatedwith line charging. This error current results from the charge that mustbe supplied to the capacitance of the transmission line. The amount ofcharging current, Icharge, increases with the length of the transmissionline. This source of error should be evaluated in situations wherein thecharging current has not been previously factored out. The fixedcovariance matrix for line charging is as follows: $\begin{matrix}{\begin{bmatrix}C_{{RR}{(n)}} & C_{{RI}{(n)}} \\C_{{IR}{(n)}} & C_{{II}{(n)}}\end{bmatrix} = {\frac{{Icharge}^{2}}{9} \cdot {( \begin{bmatrix}1 & 0 \\0 & 1\end{bmatrix} ).}}} & (37)\end{matrix}$

Another source of error is the current sensors themselves. These errorsare characterized by gain and phase angle errors as a function of themeasured current. The covariance matrix due to phase angle error iscalculated as follows: $\begin{matrix}{{\begin{bmatrix}C_{{RR}{(n)}} & C_{{RI}{(n)}} \\C_{{IR}{(n)}} & C_{{II}{(n)}}\end{bmatrix} = {\frac{\Delta \quad \varphi^{2}}{9} \cdot ( \begin{bmatrix}{PhasorImaginary}_{n}^{2} & {{- {PhasorReal}_{n}} \cdot {PhasorImaginary}_{n}} \\{{- {PhasorReal}_{n}} \cdot {PhasorImaginary}_{n}} & {PhasorReal}_{n}^{2}\end{bmatrix} )}},} & (38)\end{matrix}$

wherein Δφ is the maximum residual phase error (a design constant of arespective current sensor). The total covariance matrix for this sourceof error for each phase is the sum of the covariance matrices for eachterminal for that phase. The imaginary component of the current phasorcontributes to the real component of the covariance matrix and viceversa because a phase angle error causes an error in a phasor that isperpendicular to the phasor.

The covariance matrix due to the sensor gain error can be calculated asfollows: $\begin{matrix}{{\begin{bmatrix}C_{{RR}{(n)}} & C_{{RI}{(n)}} \\C_{{IR}{(n)}} & C_{{II}{(n)}}\end{bmatrix} = {\frac{\Delta \quad g^{2}}{9} \cdot ( \lbrack \begin{matrix}{PhasorImaginary}_{n}^{2} & {{PhasorReal}_{n} \cdot {PhasorImaginary}_{n}} \\{{PhasorReal}_{n} \cdot {PhasorImaginary}} & {PhasorReal}_{n}^{2}\end{matrix}\quad ) \rbrack}},} & (39)\end{matrix}$

wherein Δg is the maximum residual gain error (a design constant of arespective current sensor).

If the maximum residual phase and gain errors are approximately equal,the net covariance matrix for the phase and gain errors can be writtenas $\begin{matrix}{{\begin{bmatrix}C_{{RR}{(n)}} & C_{{RI}{(n)}} \\C_{{IR}{(n)}} & C_{{II}{(n)}}\end{bmatrix} = {\frac{{\Delta \quad}^{2}}{9} \cdot ( \lbrack \begin{matrix}{{PhasorReal}_{n}^{2} + {PhasorImaginary}_{n}^{2}} & 0 \\0 & {{PhasorReal}_{n}^{2} + {PhasorImaginary}_{n}^{2}}\end{matrix}\quad ) \rbrack}},} & \text{(39a)}\end{matrix}$

wherein Δ is the maximum residual error.

To provide a restraint boundary that is equivalent to a conventionalcharacteristic relating to phase and gain errors, such as a single slopepercentage restraint characteristic, the number of terminals must betaken into account when computing covariance parameters for eachterminals. The matrix can be written as follows: $\begin{matrix}{{\begin{bmatrix}C_{{RR}{(n)}} & C_{{RI}{(n)}} \\C_{{IR}{(n)}} & C_{{II}{(n)}}\end{bmatrix} = {\frac{{{slope}\quad}^{2}}{18 \cdot {terminals}} \cdot ( \lbrack \begin{matrix}{{PhasorReal}_{n}^{2} + {PhasorImaginary}_{n}^{2}} & 0 \\0 & {{PhasorReal}_{n}^{2} + {PhasorImaginary}_{n}^{2}}\end{matrix}\quad ) \rbrack}},} & \text{(39b)}\end{matrix}$

wherein slope represents the conventional percentage slope setting, andterminals represents the number of terminals in the system.

If a dual slope restraint is used, the covariance parameters due tophase and gain errors can be calculated using the following technique.First the absolute value of the phasor (PhasorAbs) is calculated:

PhasorAbs={square root over (PhasorReal²+PhasorImaginary²)}  (39c).

If PhasorAbs is less than the current at which the slope changes(Current1), then the matrix is calculated as follows: $\begin{matrix}{{\begin{bmatrix}C_{{RR}{(n)}} & C_{{RI}{(n)}} \\C_{{IR}{(n)}} & C_{{II}{(n)}}\end{bmatrix} = {\frac{{{slope1}\quad}^{2}}{18 \cdot {terminals}} \cdot ( \begin{bmatrix}{PhasorAbs}^{2} & 0 \\0 & {PhasorAbs}^{2}\end{bmatrix} )}},} & \text{(39d)}\end{matrix}$

wherein slope1 is the slope of the dual slope restraint for currentsless than Current1.

If PhasorAbs is greater than or equal to Current1, then the matrix iscalculated as follows: $\begin{matrix}{{{C_{{RI}{(n)}} = {C_{{IR}{(n)}} = 0}},{and}}\text{}{{C_{{RR}{(n)}} = {C_{{II}{(n)}} = \frac{( {{{slope1} \cdot {Current1}} + {{slope2} \cdot ( {{PhasorAbs} - {Current1}} )}} )^{2}}{18 \cdot {terminals}}}},}} & \text{(39e)}\end{matrix}$

wherein slope2 is the slope of the dual slope restraint for currentgreater than or equal to Current1.

Another potential source of error is caused by asynchronous sampling.This is a small error that arises in the computation of phasors if thenumber of data samples per cycle at the power system frequency is notexactly an integer. This error can be avoided by synchronizing samplingto the power system frequency, as discussed above.

When the each of the covariance matrices are added, the total covariancematrix defines an elliptical restraint region and can be used in thefault severity equation 36.

Transformer Protection

FIG. 6 is a circuit diagram of a transformer having windings 90 and 92with respective current sensors 94 and 96 providing current data to aprocessor 98. Current in the primarily winding in the direction of acoupling region 93 is represented by I₁ and current in the secondarywinding in the direction of the coupling region is represented by I₂^(A). Current in the secondary winding in the direction away from thecoupling region is represented by I₂ ^(B) wherein I₂ ^(A)=−I₂ ^(B).

A single phase embodiment is shown for purposes of example only;typically a three phase transformer will be used. Furthermore, althougha two winding transformer is shown, other types of multi-windingtransformers can also be used. Differential transformer protectiontechniques rely on the fact that under normal conditions the sum overall windings of the ampere turns for each winding equals the magnetizingcurrent of a transformer (usually a small quantity). Because the sum isnot identically zero, a restraint signal is needed.

Differential protection schemes work by comparing an operate signal witha restraint signal. In the present method the operate signal is derivedfrom a difference in negative sequence currents in the primary and thesecondary windings of the transformer being protected. The restraintsignal is based on an on-line calculation of the sources of measurementerror. In an adaptive procedure, as discussed above, the restraintregion is an ellipse with variable major axis, minor axis, andorientation. Parameters of the ellipse vary with time to make best useof the accuracy of current measurements.

As discussed with respect to the transmission line above, phase currentsat each winding can be measured, and decaying offsets can be removed bythe technique discussed with respect to equation 24. Then phaselets canbe calculated as discussed with respect to equations 25-27 and phasorscan be calculated as discussed with respect to equations 28-29 or 20-23.

In one embodiment, a learning stage includes the method described incommonly assigned Premerlani et al., U.S. application Ser. No.08/617,718, filed Apr. 1, 1996, wherein a function for a residualinjected negative sequence current dependent upon a positive sequencevoltage phasor and a positive sequence current phasor is determined byusing a symmetrical component transform of fundamental phasors to obtainsymmetrical component current and voltage phasors. A learning residualcurrent can be measured, for example, by subtracting a negative sequencecurrent phasor of the primary winding from a negative sequence currentof the secondary winding, and corresponding values of the positivesequence voltage and current can be monitored to determine the function.The positive sequence voltage and current of either of the two windingscan be monitored.

During operation in a protection stage, the protection residual currentcan also be determined by calculating and subtracting a negativesequence current phasor of one winding in the direction away fromcoupling region 93 from a negative sequence current phasor of the otherwinding in the direction of the coupling region (or, equivalently,adding the two negative sequence current phasors in the direction of thecoupling region). A corresponding learning residual current phasor canbe subtracted from the protection residual current phasor with theresulting phasor being used below in comparison with the ellipticalrestraint region.

When transformers having more than two windings are used, the protectionresidual current can be determined by adding all the negative sequencecurrent phasors in the direction of coupling region 93.

During operation, phaselet sums can be determined from current samplesat each winding, and the elliptical restraint region from sources ofmeasurement error can be calculated using the above equations 34 and37-39 relating to the covariance matrix, thus effectively applying totalharmonic restraint for inrush and overexcitation in transformerprotection. If the resulting phasor falls within the restraint region,no fault is present. If the resulting phasor falls outside the restraintregion, a fault is present. In the unlikely event that the resultingphasor falls on the very thin boundary of the restraint region, whethera fault is present is uncertain. A fault can arbitrarily be declared inthis situation. If a fault is present or declared, a filter process canbe used to determine whether to trip the circuit.

While only certain preferred features of the invention have beenillustrated and described herein, many modifications and changes willoccur to those skilled in the art. It is, therefore, to be understoodthat the appended claims are intended to cover all such modificationsand changes as fall within the true spirit of the invention.

What is claimed is:
 1. A method of consolidating phase current samplesfor transmission from one location to another location, the methodincluding: at the one location, obtaining the phase current samples, andcalculating real and imaginary phaselets comprising partial sums of thephase current samples; transmitting the real and imaginary phaseletsfrom the one location to the another location; and at the anotherlocation, calculating sums of the real and imaginary phaselets over avariable size sliding sample window, and calculating real and imaginaryphasor components over the sample window by multiplying the sums of thereal and imaginary phaselets by a phaselet transformation matrix.
 2. Themethod of claim 1, wherein the phaselet transformation matrix comprises:$\begin{matrix}{\begin{bmatrix}{T_{RR}( {n,W} )} & {T_{RI}( {n,W} )} \\{T_{IR}( {n,W} )} & {T_{II}( {n,W} )}\end{bmatrix},\quad {where}} \\\begin{matrix}{T_{RR} = \quad {( {1 - ( {\frac{1}{W} \cdot {\cos ( {\frac{4 \cdot \pi \cdot P \cdot n}{N} - \frac{2 \cdot \pi \cdot W}{N}} )} \cdot \frac{\sin ( \frac{2 \cdot \pi \cdot W}{N} )}{\sin ( \frac{2 \cdot \pi}{N} )}} )} ) \cdot}} \\{\quad \frac{2}{W - \frac{( {\sin ( \frac{2 \cdot \pi \cdot W}{N} )} )^{2}}{W \cdot ( {\sin ( \frac{2 \cdot \pi}{N} )} )^{2}}}}\end{matrix} \\\begin{matrix}{T_{RI} = \quad {T_{IR} = {\frac{1}{W} \cdot {\sin ( {\frac{4 \cdot \pi \cdot P \cdot n}{N} - \frac{2 \cdot \pi \cdot W}{N}} )} \cdot \frac{\sin ( \frac{2 \cdot \pi \cdot W}{N} )}{\sin ( \frac{2 \cdot \pi}{N} )} \cdot}}} \\{\quad \frac{2}{W - \frac{( {\sin ( \frac{2 \cdot \pi \cdot W}{N} )} )^{2}}{W \cdot ( {\sin ( \frac{2 \cdot \pi}{N} )} )^{2}}}}\end{matrix} \\{and} \\\begin{matrix}{T_{II} = \quad {( {1 + ( {\frac{1}{W} \cdot {\cos ( {\frac{4 \cdot \pi \cdot P \cdot n}{N} - \frac{2 \cdot \pi \cdot W}{N}} )} \cdot \frac{\sin ( \frac{2 \cdot \pi \cdot W}{N} )}{\sin ( \frac{2 \cdot \pi}{N} )}} )} ) \cdot}} \\{\quad \frac{2}{W - \frac{( {\sin ( \frac{2 \cdot \pi \cdot W}{N} )} )^{2}}{W \cdot ( {\sin ( \frac{2 \cdot \pi}{N} )} )^{2}}}}\end{matrix}\end{matrix}$

and wherein W represent a number of samples in the sample window, Nrepresents a number of samples in a sample cycle, and n represents aphasor index number.
 3. The method of claim 1, further including foreach phaselet, calculating a respective partial sum of squares of eachphase current sample; calculating a sum of the partial sums of thesquares over the sample window; using the sums of the real and imaginaryphaselets, the real and imaginary phasor components, and the sum of thepartial sums of the squares to calculate a sum of squares of errorsbetween the phase current samples and a fitted sine wave representativeof the real and imaginary phasor components; using the sum of squares oferrors to calculate a variance matrix defining an elliptical uncertaintyregion; a determining whether a disturbance has occurred, and, if so,re-initializing the sample window; and determining whether a sum ofcurrent phasors for a respective phase falls outside of the ellipticaluncertainty region for the respective phase.
 4. The method of claim 3,wherein the step of using the sums of the real and imaginary phaselets,the real and imaginary phasor components, and the sum of the partialsums of the squares to calculate a sum of squares of errors between thephase current samples and a fitted sine wave representative of the realand imaginary phasor components comprises subtracting from the sum ofthe partial sums of the squares, the product of the real phasorcomponent and the sum of the real phaselets and the product of theimaginary phasor component and the sum of the imaginary phaselets. 5.The method of claim 4, wherein the step of determining whether adisturbance has occurred includes monitoring at least one of the sum ofsquares of errors; a magnitude of a positive sequence current, anegative sequence current, or a ground current; a change in a positive,a negative, or a zero sequence current; or a change in load current. 6.The method of claim 4, wherein the variance matrix comprises thefollowing equation: $\begin{bmatrix}C_{{RR}{({n,W})}} & C_{{RI}{({n,W})}} \\C_{{IR}{({n,W})}} & C_{{II}{({n,W})}}\end{bmatrix} = {\frac{E_{n}^{2}}{W} \cdot ( \begin{bmatrix}T_{{RR}{({n,W})}} & T_{{RI}{({n,W})}} \\T_{{IR}{({n,W})}} & T_{{II}{({n,W})}}\end{bmatrix} )}$

wherein C_(RR) represents an expected value of a square of an error inthe real phasor component, C_(RI) and C_(IR) represent an expected valueof a product of errors in the real and imaginary phasor components,C_(II) represents an expected value of a square of an error in theimaginary phasor component, E_(n) ² represents the sum of squares oferrors, W represents a number of samples in the sample window, nrepresents a phasor index, and T_(RR(n,W)), T_(RI(n,W)), T_(IR(n, W)),and T_(II(n,W)) represent a phaselet transformation matrix.
 7. A systemfor consolidating phase current samples for transmission from onelocation to another location, the system including: at the one location,current sensors for obtaining the phase current samples, and a firstcomputer for calculating real and imaginary phaselets comprising partialsums of the phase current samples; a transmission line for transmittingthe real and imaginary phaselets from the one location to the anotherlocation; and at the another location, a second computer for calculatingsums of the real and imaginary phaselets over a variable size slidingsample window, and calculating real and imaginary phasor components overthe sample window by multiplying the sums of the real and imaginaryphaselets by a phaselet transformation matrix.
 8. A method ofconsolidating phase current samples for transmission from one locationto another location, the method comprising: at the one location,obtaining the phase current samples, and calculating real and imaginaryphaselets comprising partial sums of the phase current samples;transmitting the real and imaginary phaselets from the one location tothe another location; at the another location, calculating sums of thereal and imaginary phaselets over a variable size sliding sample window,calculating real and imaginary phasor components over the sample windowby multiplying the sums of the real and imaginary phaselets by aphaselet transformation matrix, for each phaselet, calculating arespective partial sum of squares of each phase current sample;calculating a sum of the partial sums of the squares over the samplewindow; using the sums of the real and imaginary phaselets, the real andimaginary phasor components, and the sum of the partial sums of thesquares to calculate a sum of squares of errors between the phasecurrent samples and a fitted sine wave representative of the real andimaginary phasor components; using the sum of squares of errors tocalculate a variance matrix defining an elliptical uncertainty region;determining whether a disturbance has occurred, and, if so,re-initializing the sample window; and determining whether a sum ofcurrent phasors for a respective phase falls outside of the ellipticaluncertainty region for the respective phase.
 9. The method of claim 8,wherein the step of using the sums of the real and imaginary phaselets,the real and imaginary phasor components, and the sum of the partialsums of the squares to calculate a sum of squares of errors between thephase current samples and a fitted sine wave representative of the realand imaginary phasor components comprises subtracting from the sum ofthe partial sums of the squares, the product of the real phasorcomponent and the sum of the real phaselets and the product of theimaginary phasor component and the sum of the imaginary phaselets. 10.The method of claim 9, wherein the step of determining whether adisturbance has occurred includes monitoring at least one of the sum ofsquares of errors; a magnitude of a positive sequence current, anegative sequence current, or a ground current; a change in a positive,a negative, or a zero sequence current; or a change in load current. 11.A system for consolidating phase current samples for transmission fromone location to another location, the system comprising: at the onelocation, means for obtaining the phase current samples, and calculatingreal and imaginary phaselets comprising partial sums of the phasecurrent samples; means for transmitting the real and imaginary phaseletsfrom the one location to the another location; and at the anotherlocation, means for calculating sums of the real and imaginary phaseletsover a variable size sliding sample window, and calculating real andimaginary phasor components over the sample window by multiplying thesums of the real and imaginary phaselets by a phaselet transformationmatrix.
 12. The system of claim 7, wherein the phaselet transformationmatrix comprises: $\begin{matrix}{\begin{bmatrix}{T_{RR}( {n,W} )} & {T_{RI}( {n,W} )} \\{T_{IR}( {n,W} )} & {T_{II}( {n,W} )}\end{bmatrix},\quad {where}} \\\begin{matrix}{T_{RR} = \quad {( {1 - ( {\frac{1}{W} \cdot {\cos ( {\frac{4 \cdot \pi \cdot P \cdot n}{N} - \frac{2 \cdot \pi \cdot W}{N}} )} \cdot \frac{\sin ( \frac{2 \cdot \pi \cdot W}{N} )}{\sin ( \frac{2 \cdot \pi}{N} )}} )} ) \cdot}} \\{\quad \frac{2}{W - \frac{( {\sin ( \frac{2 \cdot \pi \cdot W}{N} )} )^{2}}{W \cdot ( {\sin ( \frac{2 \cdot \pi}{N} )} )^{2}}}}\end{matrix} \\\begin{matrix}{T_{RI} = \quad {T_{IR} = {\frac{1}{W} \cdot {\sin ( {\frac{4 \cdot \pi \cdot P \cdot n}{N} - \frac{2 \cdot \pi \cdot W}{N}} )} \cdot \frac{\sin ( \frac{2 \cdot \pi \cdot W}{N} )}{\sin ( \frac{2 \cdot \pi}{N} )} \cdot}}} \\{\quad \frac{2}{W - \frac{( {\sin ( \frac{2 \cdot \pi \cdot W}{N} )} )^{2}}{W \cdot ( {\sin ( \frac{2 \cdot \pi}{N} )} )^{2}}}}\end{matrix} \\{and} \\\begin{matrix}{T_{II} = \quad {( {1 + ( {\frac{1}{W} \cdot {\cos ( {\frac{4 \cdot \pi \cdot P \cdot n}{N} - \frac{2 \cdot \pi \cdot W}{N}} )} \cdot \frac{\sin ( \frac{2 \cdot \pi \cdot W}{N} )}{\sin ( \frac{2 \cdot \pi}{N} )}} )} ) \cdot}} \\{\quad \frac{2}{W - \frac{( {\sin ( \frac{2 \cdot \pi \cdot W}{N} )} )^{2}}{W \cdot ( {\sin ( \frac{2 \cdot \pi}{N} )} )^{2}}}}\end{matrix}\end{matrix}$

and wherein W represents a number of samples in the sample window, Nrepresents a number of samples in a sample cycle, and n represents aphasor index number.
 13. The system of claim 7, wherein the secondcomputer is further adapted to for each phaselet, calculate a respectivepartial sum of squares of each phase current sample; calculate a sum ofthe partial sums of the squares over the sample window; use the sums ofthe real and imaginary phaselets, the real and imaginary phasorcomponents, and the sum of the partial sums of the squares to calculatea sum of squares of errors between the phase current samples and afitted sine wave representative of the real and imaginary phasorcomponents; use the sum of squares of errors to calculate a variancematrix defining an elliptical uncertainty region; determine whether adisturbance has occurred, and, if so, re-initialize the sample window;and determine whether a sum of current phasors for a respective phasefalls outside of the elliptical uncertainty region for the respectivephase.
 14. The system of claim 13, wherein the second computer isadapted to use the sums of the real and imaginary phaselets, the realand imaginary phasor components, and the sum of the partial sums of thesquares to calculate a sum of squares of errors between the phasecurrent samples and a fitted sine wave representative of the real andimaginary phasor components by subtracting from the sum of the partialsums of the squares, the product of the real phasor component and thesum of the real phaselets and the product of the imaginary phasorcomponent and the sum of the imaginary phaselets.
 15. The system ofclaim 14, wherein the second computer is adapted to determine whetherthe disturbance has occurred by monitoring at least one of the sum ofsquares of errors; a magnitude of a positive sequence current, anegative sequence current, or a ground current; a change in a positive,a negative, or a zero sequence current; or a change in load current. 16.The system of claim 14, wherein the variance matrix comprises thefollowing equation: $\begin{bmatrix}C_{{RR}{({n,W})}} & C_{{RI}{({n,W})}} \\C_{{IR}{({n,W})}} & C_{{II}{({n,W})}}\end{bmatrix} = {\frac{E_{n}^{2}}{W} \cdot ( \begin{bmatrix}T_{{RR}{({n,W})}} & T_{{RI}{({n,W})}} \\T_{{IR}{({n,W})}} & T_{{II}{({n,W})}}\end{bmatrix} )}$

wherein C_(RR) represents an expected value of a square of an error inthe real phasor component, C_(RI) and C_(IR) represent an expected valueof a product of errors in the real and imaginary phasor components,C_(II) represents an expected value of a square of an error in theimaginary phasor component, E_(n) ² represents the sum of squares oferrors, W represents a number of samples in the sample window, nrepresents a phasor index, and T_(RR(n,W)), T_(RI(n,W)), T_(IR(n,W)),and T_(II(n,W)) represent a phaselet transformation matrix.